Kinematical X-ray Diffraction Research Articles

Dynamical and kinematical X-ray diffraction in a bent crystal

Numerical modeling of kinematical and dynamical X-ray diffraction in a bent crystal was performed on the basis of two approaches to integrating the Takagi–Taupin equations, and using two-dimensional recurrence relations. Within the framework of kinematical diffraction, a new equation is obtained that describes the distribution of diffracted intensity inside a bent crystal. The time taken for numerical calculations based on this equation is significantly reduced in comparison with the use of algorithms of the dynamical diffraction theory. The simulation shows for the first time that, for strongly bent crystals, the maximum value of the diffraction intensity is formed inside the deformed structure and not on its surface. In the case of strong bending of the crystal structure, the deviation of the X-ray beam from the Bragg angle does not change the diffraction pattern but shifts it along the lateral direction. The results of calculations of diffraction in a strongly bent crystal based on the equations of dynamical and kinematical diffraction coincide, while the computations for weakly bent crystals differ. The possibility of estimating the primary extinction length of a bent crystal as a function of the bending radius is shown. In the case of kinematical diffraction in bent crystalline microsystems, a new method has been developed to calculate X-ray reciprocal-space mapping.

Solving the crystallographic phase problem using dynamical scattering in electron diffraction

Solving crystal structures from kinematical X-ray or electron diffraction patterns of single crystals requires many more diffracted beams to be recorded than there are atoms in the structure, since the phases of the structure factors can only be retrieved from such data if the atoms can be resolved as sharply peaked objects. Here a method is presented by which the fact that multiple scattering encodes structure factor phases in the diffracted intensities is being used for solving the crystallographic phase problem. The retrieval of both amplitudes and phases of electron structure factors from diffraction patterns recorded with varying angle of incidence will be demonstrated. No assumption about the scattering potential itself is being made. In particular, the resolution in the diffraction data does not need to be sufficient to resolve atoms, making this method potentially interesting for electron crystallography of 2-dimensional protein crystals and other beam-sensitive complex structures.

Diffuse X-ray scattering from local chemical inhomogeneities in InGaN layers

Diffuse X-ray scattering from random chemical inhomogeneities in epitaxial layers of InGaN/GaN was simulated using linear elasticity theory and kinematical X-ray diffraction. The simulation results show the possibility of determining the r.m.s. deviations of the local In content and its lateral correlation length from reciprocal-space maps of the scattered intensity. The reciprocal-space distribution of the intensity scattered from inhomogeneities is typical and it can be distinguished from other sources of diffuse scattering such as threading or misfit dislocations.

CADEM: calculate X-ray diffraction of epitaxial multilayers.

Epitaxial multilayers and superlattice (SL) structures are gaining increasing importance as they offer the opportunity to create artificial crystals with new functionalities. These crystals deviate from the parent bulk compounds not only in terms of the lattice constants but also in the symmetry classification, which renders calculation of their X-ray diffraction (XRD) patterns tedious. Nevertheless, XRD is essential to get information on the multilayer/SL structure such as, for example,out-of-plane lattice constants, strain relaxation and period length of the crystalline SL. This article presents a powerful yet simple program, based on the general one-dimensional kinematic X-ray diffraction theory, which calculates the XRD patterns of tailor-made multilayers and thus enables quantitative comparison of measured and calculated XRD data. As the multilayers are constructed layer by layer, the final material stack can be entirely arbitrary. Moreover, CADEM is very flexible and can be straightforwardly adapted to any material system. The source code of CADEM is available as supporting material for this article.

Paracrystalline model in statistical theory of X-ray diffraction on epilayers with quantum dots

A statistical theory of kinematical X-ray diffraction on nanostructured crystalline systems with quantum dots (QDs) has been developed with allowance for a spatial correlation of QDs. The lateral distribution of QDs in an epilayer is described using a two-dimensional paracrystalline model. The angular distribution of diffuse X-ray scattering intensity was numerically simulated and analyzed depending on the degree of lateral ordering of QDs.

Concentration and relaxation depth profiles of InxGa1−xAs/GaAs and GaAs1−xPx/GaAs graded epitaxial films studied by x-ray diffraction

A method is proposed to determine the concentration and relaxation depth profiles in graded epitaxial films from x-ray reciprocal space maps (RSMs). Various approximations in the kinematical x-ray diffraction from epitaxial films with the misfit dislocation density depth profile are developed. We show that a symmetric and an asymmetric RSM, or two asymmetric RSMs, contain enough information to obtain the concentration, relaxation, and lattice tilt depth profiles without any additional assumptions. The proposed approach is applied to In${}_{x}$Ga${}_{1\ensuremath{-}x}$As/GaAs and GaAs${}_{1\ensuremath{-}x}$P${}_{x}$/GaAs epitaxial graded films. The reconstructed concentration and dislocation density depth profiles are found to be in an agreement with the ones expected from the growth conditions.

Modelling of diffraction from fibre texture gradients in thin polycrystalline films

Crystallographic textures in thin polycrystalline films typically exhibit a rotational symmetry,i.e.they occur as a fibre texture with the texture pole being orientated in the direction of the substrate normal. As a further characteristic of thin-film textures, it was often observed that the degree of preferred orientation increases with increasing thickness. It is shown in this work how a fibre texture gradient may be modelled in kinematical X-ray diffraction and which effects it has on the intensity mapping of theIHKLreflection, when theHKLpole is the fibre axis. A general expression forIHKLis derived for a depth-dependent fibre texture that is based on the finite Laplace transform of the texture distribution. The concept is outlined for the cosnψ function to model the tilt-angle dependence of intensity, with the parameterndenoting the degree of texture. It is found that the measured intensity distribution sensitively depends on the ratio of texture gradient over X-ray attenuation coefficient. For particular cases, it is found that the maximum intensity may occur for non-zero tilt angles and thus arise at a different tilt angle from the pole of the fibre texture.

Form and structure factors for impedance and reflection from periodic layers

In an exact treatment of the Maxwell equations, we derive form and structure factors for reflection from periodic layers, and we show that these factors are significantly different from their analogs in kinematic x-ray diffraction. Quite generally, we show that reflection and impedance can be written precisely as the sum of an additive form factor and the product of a structure factor and a second form factor. This additive form factor does not have an analog in kinematic x-ray diffraction. It is demonstrated that the form factors are found by analytic continuation to an arbitrary wavelength of expressions for the impedance both at long wavelengths and at quarter wavelengths. A correction to the Bragg law relating fringe spacing to the total structure thickness is derived. We go beyond previous numerical work by deriving simple analytic exact expressions for reflection and impedance of periodic layers for all frequencies within the reflection passband, and for an arbitrary number of periods in the structure, an arbitrary index profile within each period, arbitrary layer thicknesses (not just quarter-wave layers), and for arbitrary sizes of the refractive index differences.

The statistical kinematical theory of X-ray diffraction as applied to reciprocal-space mapping

The statistical kinematical X-ray diffraction theory is developed to describe reciprocal-space maps (RSMs) from deformed crystals with defects of the structure. The general solutions for coherent and diffuse components of the scattered intensity in reciprocal space are derived. As an example, the explicit expressions for intensity distributions in the case of spherical defects and of a mosaic crystal were obtained. The theory takes into account the instrumental function of the triple-crystal diffractometer and can therefore be used for experimental data analysis.

Growth Twins in YbxY1−xAl3(BO3)4 and NdxGd1−xAl(BO3)4 Crystals Observed by White-Beam Synchrotron Radiation Topography

ABSTRACTGrowth twins in Ybx Y1−xAl3(BO3)4 (YbYAB) and NdxGd1−xAl3(BO3)4 (NGAB) crystals were observed by white-beam synchrotron radiation topography combined with chemical etch. It was found that growth twins in YbYAB crystals are of inversion types in which two twinned pairs have the central inversion relationship. This kind of twinning was visible in x-ray topography not by ‘domain contrast’ but by ‘boundary contrast’ stemming from the kinematical x-ray diffraction at the boundary. Growth twins in NGAB crystals are of 1800 rotation types in which the twofold symmetric operation axis is parallel to the [0001] axis. This kind of twinning often shows black-and-white contrast in x-ray topography which originates from the different structure factors between twinned pairs. In addition. the formation mechanisms of growth twins are discussed.

X-ray crystal truncation rod scattering analysis of reactive ion etched silicon

We have investigated lattice damage in reactive ion etched silicon by using x-ray crystal truncation rod (CTR) scattering. The x-ray intensity associated with the rod in the reciprocal space depends on the etching-induced lattice distortion. To estimate the magnitude of the lattice distortion, we analyzed the obtained data with a kinematic x-ray diffraction model on the assumption that the lattice distortion decays exponentially with the depth. We found that the lattice distortion extends to a depth of about 9 nm. In addition, we propose a method for quantitatively evaluating lattice damage based on our analysis. This method allows us to compare lattice damage among samples etched under different conditions. This study indicates that x-ray CTR experiments provide a useful means of characterizing lattice distortions near processed surfaces.

Time-dependent X-ray Bragg diffraction

The theory of time-dependent X-ray Bragg diffraction by crystals is developed on the basis of the Green-function (point-source) formalism. A general case of incident radiation partially coherent in time and space is considered. The time-delay effect of the diffracted radiation is described when the ultrashort time duration incident pulse strikes the crystal surface. The problem in question is closely connected with the effect of time delay in the resonance scattering of synchrotron radiation by nuclei in a crystal. It is found that, for the case where the incident wave is plane (or is an incoherent superposition of plane waves) and the time-dependent pulse is a pseudo δ function in time, the instantaneous crystal reflectivity is a smooth temporal function and tends to the value corresponding to the integrated reflectivity calculated by means of the conventional dynamical–kinematical X-ray diffraction theory. If the incident X-ray pulse profile is a pseudo 6 function in both time and space, the temporal crystal response has the same functional dependence as the spatial distribution of the diffracted intensity under the condition of conventional Bragg diffraction of the X-ray beam with lateral width < toc, where the time delay to is equal to Λ/2πc and (μoC)−1 in the cases of dynamical and kinematical X-ray scattering within a crystal, respectively (Λ is the X-ray extinction length, μo is the linear absorption coefficient and c is the velocity of light in vacuum).

Structural investigation of II–VI compound semiconductor quantum wires using triple-axis X-ray diffractometry

Corrugated CdZnTe substrates and quantum wires fabricated by dry etching techniques from (CdTe)/(MnTe) superlattices (SLs), which were grown by a combination of molecular beam and atomic layer epitaxy, were investigated with respect to their structural properties. The SLs were grown pseudomorphically on Cd 0.96Zn 0.04Te (001) oriented substrates. X-ray reciprocal space mapping around (004) and (115) reciprocal lattice points was used to measure the periods both along the growth and the lateral corrugation direction. The arrays were defined by holographic lithography and reactive ion etching with a typical period of about 400 nm. For the analysis of the corrugations, a kinematical X-ray diffraction model is used which yields their geometrical shape and also informations on the crystalline damage. Furthermore, it turns out that the reactive ion etching process causes an increase of the mean SL lattice constant of about 0.1% to 0.15% along the growth direction.

Superconducting Multilayers: Microstructural Properties Studied by X-Ray Diffraction

ABSTRACTStructural properties of molecular beam epitaxy-grown Bi2Sr2Cu1Oy'Bi2Sr2Ca1Cu2Oy multilayers have been studied by x-ray diffraction. A one-dimensional kinematic x-ray diffraction model has been used to describe the structural quality of the multilayers. Interface roughness, stacking defects and unit cell disorder are obtained by an iterative fitting of the calculated diffraction profile to the experimental spectra. The type and amount of disorder in the films was qualitatively determined. Results demonstrate that structural imperfections are present in the multilayers and have to be considered when transport properties are studied.

The measurement of defect parameters in imperfect crystals using X-ray diffraction

Correlation lengths and defect-strength parameters, related to the separations and magnitudes of discontinuities in imperfect crystals, are obtained from X-ray rocking curves using a stochastic model of crystal defects. The model describes the diffraction of X-rays from an imperfect crystal containing surfaces of defects, such as stacking faults, and misoriented crystal grains. The two defect parameters provide a measure of crystal quality. A method of extracting the parameters from rocking curves is described in the limit of kinematic X-ray diffraction. The method is applied to X-ray diffraction data obtained from thin films of CdTe and Hg 1-x Cd x Te grown on GaAs substrates. The ability of the model to fit the X-ray data is a test of the stochastic model

X-ray characterization of Au/Ni multilayer thin films

Au/Ni multilayer thin films of single phase structure were analyzed using kinematic X-ray diffraction theories to determine the composition modulation and structure of this multilayer system. The repeat period of the multilayers used in this study ranged from 0.82 × 10 −9m to 4.26 × 10 −9m. The composition modulation was found to be a near square wave. The strain at the AuNi interface is predicted to be 9.8% at repeat period 1.65 × 10 −9m with decreasing values above and below this repeat period.

X-ray standing waves in the Laue case – location of impurity atoms

Impurity-atom fluorescence excited by X-ray standing waves in the Laue case of X-ray diffraction has been investigated experimentally and theoretically. Possibilities for location of impurity atoms in the bulk and the surface layer of single crystals are discussed. The experiments were carried out on silicon crystals of different thicknesses doped with germanium. The general approach for calculation of the fluorescence-yield angular curves has been developed. In the case of the uniform distribution of impurity atoms in the bulk of a crystal and also in the case of the kinematical X-ray diffraction on a thin surface layer, analytical expressions can be used.

Improved Kematical X-Ray Rocking Cjrve Analyses

ABSTRACTA new boundary condition is used in the kinematical x-ray diffraction model analysis of rocking curves. The boundary condition is the dynamical reflection amplitude instead of the previously used dynamical intensity for the substrate. It is shown that this boundary condition properly accounts for the angular shift effect in the Bragg peak profile of very thin epitaxial layers and in the 0th-order superlattice peak. We present experimental and simulated rocking curves for various samples. The simulation was performed by using the dynamical diffraction theory, the kinematical model with the new amplitude boundary condition, and the kinematical model with the old intensity boundary condition. We also analyzed the strained resonant tunneling device structure and single strained quantum well samples by x-ray interference technique. Our x-ray interference results showed a good agreement with the nominal values for both experimental and simulated rocking curves.

On the transition between dynamical and kinematical X-ray diffraction in thin crystals with randomly distributed dislocations

physica status solidi (a)Volume 116, Issue 2 p. K141-K145 Short Note On the transition between dynamical and kinematical X-ray diffraction in thin crystals with randomly distributed dislocations V. I. Khrupa, V. I. Khrupa Institute of Semiconductors, Academy of Sciences of the Ukrainian SSR, Kiev Search for more papers by this authorV. V. Nikolaev, V. V. Nikolaev Institute of Semiconductors, Academy of Sciences of the Ukrainian SSR, Kiev Search for more papers by this authorM. Ya. Skorokhod, M. Ya. Skorokhod Institute of Semiconductors, Academy of Sciences of the Ukrainian SSR, Kiev Search for more papers by this author V. I. Khrupa, V. I. Khrupa Institute of Semiconductors, Academy of Sciences of the Ukrainian SSR, Kiev Search for more papers by this authorV. V. Nikolaev, V. V. Nikolaev Institute of Semiconductors, Academy of Sciences of the Ukrainian SSR, Kiev Search for more papers by this authorM. Ya. Skorokhod, M. Ya. Skorokhod Institute of Semiconductors, Academy of Sciences of the Ukrainian SSR, Kiev Search for more papers by this author First published: 16 December 1989 https://doi.org/10.1002/pssa.2211160248Citations: 2 Prospekt Nauki 45, SU-252028 Kiev, USSR. AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volume116, Issue216 December 1989Pages K141-K145 RelatedInformation

Rocking-Curve Peak Shift in Thin Heterojunction Single Layers

AbstractA simple method for determining layer composition and mismatch of semiconductor hetero-epitaxial samples is by measuring the separation of peaks in x-ray rocking curve (XRC). This method fails if the peak separation is affected by other factors. For a small layer thickness, the layer peak position is affected by the x-ray amplitudes of the substrate or other thicker layers through the interference and overlap effects. In this case, a diffraction theory fitting process is necessary for a correct determination of layer parameters. We have used dynamical and kinematical x-ray diffraction theories to calculate the layer peak position as a function of its thickness for various layer/substrate combinations. These two theories yield substantially different results, indicating that the kinematical diffraction theory analysis is no longer valid for these thin layers. When a thick layer is present along with the thin layer, the thick layer is more influential than the substrate to the thin layer peak position, making the dynamical theory fitting necessary even from higher thickness.